Modeling the Dispersion of the Nonlinearity in Slow Mode Photonic Crystal Waveguides

Nanophotonic structures enabling the engineering of linear dispersion are often exploited for nonlinear processing. However, the dispersion of the nonlinear response has been generally overlooked, also because of the difficulty of accurately taking this effect into account. Here, we first show the necessity of considering the nonlinear dispersion and then propose a simplified approach through which the modeling can be performed. As an example, we consider waveguides in which a large group index and low group velocity dispersion are achieved at the same time by modifying a single design parameter, i.e., an asymmetric translation of the hole rows closest to the core. It is shown that the dependence on the wavelength of the normalized Bloch mode overlap integrals for self-phase modulation (SPM) can be interpolated by a four-parameter formula that is similar to the Morse potential. The parameters can be related to specific properties of the nonlinear SPM coefficient, which in the specific case depends on the translation parameter, by simple linear and exponential functions. Cross-phase modulation and four-wave mixing coefficients can then be derived from the self-phase one, through a geometric mean, slightly corrected to account for the interacting wave detuning, and the complete modeling of nonlinear dispersion is achieved.